Curry kept the Warriors in the game down the stretch, netting eight of his total in the fourth quarter and six more in extra time as Golden State snapped the Rockets' eight-game winning streak.
David Lee had 28 points and 14 rebounds.
Andre Iguodala helped out with 11 points, eight rebounds and seven assists for the Warriors.
Houston fell despite 39 points from James Harden.
Chandler Parsons contributed 21 points, eight rebounds and five assists.